Connie, I just pinned the pieces to the background and satin stitched around them. A satin stitch is usually one of your decorative stitches on your sewing machine. On mine it looks like a zig zag, but when you shorten the stitch length and loosen the tension a bit it comes out close
I did not use a stabilizer. Look in your instruction manual, they explain how to use the stitches and then try out the different lengths until you find what you want.
Mary